Abstract
A Bayesian design is given by maximising an expected utility over a design space. The utility is chosen to represent the aim of the experiment and its expectation is taken with respect to all unknowns: responses, parameters and/or models. Although straightforward in principle, there are several challenges to finding Bayesian designs in practice. Firstly, the utility and expected utility are rarely available in closed form and require approximation. Secondly, the design space can be of high-dimensionality. In the case of intractable likelihood models, these problems are compounded by the fact that the likelihood function, whose evaluation is required to approximate the expected utility, is not available in closed form. A strategy is proposed to find Bayesian designs for intractable likelihood models. It relies on the development of an automatic, auxiliary modelling approach, using multivariate Gaussian process emulators, to approximate the likelihood function. This is then combined with a copula-based approach to approximate the marginal likelihood (a quantity commonly required to evaluate many utility functions). These approximations are demonstrated on examples of stochastic process models involving experimental aims of both parameter estimation and model comparison.
Highlights
Often, the dynamics underpinning a complex physical phenomenon can be modelled by a stochastic process
We initially describe the concept of Bayesian optimal design of experiments for the experimental aim of parameter estimation
In this paper we have introduced a general-purpose approach for finding Bayesian designs under intractable likelihood models
Summary
The dynamics underpinning a complex physical phenomenon can be modelled by a stochastic process. It is commonly the situation that the stochastic process (or model) depends on unknown parameters, time and, potentially, other controllable variables. We consider the case where an experiment is to be performed to learn about the phenomenon by estimating the unknown parameters. The physical phenomenon of interest is observed at a series of time points, after the specification of any controllable variables, and the stochastic model is fitted to the observed responses. N, the kth run involves the specification of a w × 1 vector of design variables dk ∈ D. Where F is a distribution depending on a p × 1 vector of unknown parameters θ ∈ Θ, with Θ the parameter space. The pdf/pmf of this distribution is called the marginal likelihood ( known as evidence) and given by π(y|D) = π(y|θ, D)π(θ)dθ,
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